Weak formulation

Weak formulations are an important tool for the analysis of mathematical equations that permit the transfer of concepts of linear algebra to solve problems in other fields such as partial differential equations. In a weak formulation, an equation is no longer required to hold absolutely (and this is not even well defined) and has instead weak solutions only with respect to certain "test vectors" or "test functions". This is equivalent to formulating the problem to require a solution in the sense of a distribution.

We introduce weak formulations by a few examples and present the main theorem for the solution, the Lax–Milgram theorem.

This is what is usually called the weak formulation of Poisson's equation; what's missing is the space , which is beyond the scope of this article. The space must allow us to write down this equation. Therefore, we should require that the derivatives of functions in this space are square integrable. Now, there is actually the Sobolev space of functions with weak derivatives in and with zero boundary conditions, which fulfills this purpose.